# Thought experiments that exact hat algebra can and cannot compute

Among other things, I’m teaching the Eaton-Kortum (2002) model and “exact hat algebra” to my PhD class tomorrow. Last year, my slides said “this model’s counterfactual predictions can be obtained without knowing all parameter values by a procedure that we now call ‘exact hat algebra’.” Not anymore. Only some of its counterfactual predictions can be attained via that technique.

As I reviewed in a 2018 blog post, when considering a counterfactual change in trade costs (and no change in exogenous productivities nor population sizes), the exact-hat-algebra calculation requires only the trade elasticity and initial trade flows in order to solve for the endogenous proportionate wage changes associated with any choice of exogenous proportionate trade-cost changes.

In Section 6.1 of Eaton and Kortum (2002), the authors consider two counterfactual scenarios that speak to the gains from trade. The first raises trade costs to their autarkic levels (“dni goes to infinity”). The second eliminates trade costs (“dni goes to one”). Exact hat algebra can be used to compute the first counterfactual; see Costinot and Rodriguez-Clare (2014) for a now-familiar exposition or footnote 42 in EK (setting α = β = 1). The second counterfactual cannot be computed by exact hat algebra.

One cannot compute the “zero-gravity” counterfactual of Eaton and Kortum (2002) using exact hat algebra because this would require one to know the initial levels of trade costs. To compute the proportionate change in trade costs associated with the dni=1 counterfactual, one would need to know the values of the “factual” dni. The exact hat algebra procedure doesn’t identify these values. Exact hat algebra allows one to compute proportionate changes in endogenous prices in an underidentified model by leveraging implicit combinations of parameter values that rationalize the observed initial equilibrium without separately identifying them.

Exact hat algebra requires only the trade elasticity and the initial trade matrix (including expenditures on domestically produced goods). That’s not enough to identify the model’s parameters. (If these moments alone sufficed to identify bilateral trade costs, the Head-Ries index that only computes their geometric mean wouldn’t be necessary.) Thus, one can only use exact hat algebra to compute outcomes for counterfactual scenarios that don’t require full knowledge of the model’s parameter values. One can express the autarky counterfactual in proportionate changes (“d-hat is infinity”), but one cannot define the proportionate change in trade costs for the “zero-gravity” counterfactual without knowing the initial levels of trade costs. There are some thought experiments that exact hat algebra cannot compute.

Update (5 Oct): My comment about the contrast between the two counterfactuals in section 6.1 of Eaton and Kortum (2002) turns out to be closely related to the main message of Waugh and Ravikumar (2016). They and Eaton, Kortum, Neiman (2016) both show ways to compute the frictionless or “zero-gravity” equilibria when using additional data (namely, prices or price deflators). See also footnote 7 of Sposi, Santacreu, Ravikumar (2019), which is attached to the sentence “Note that reductions of trade costs (dij − 1) require knowing the initial value of dij.”

# Spatial Economics for Granular Settings

Economists studying spatial connections are excited about a growing body of increasingly fine spatial data. We’re no longer studying country- or city-level aggregates. For example, many folks now leverage satellite data, so that their unit of observation is a pixel, which can be as small as only 30 meters wide. See Donaldson and Storeygaard’s “The View from Above: Applications of Satellite Data in Economics“. Standard administrative data sources like the LEHD publish neighborhood-to-neighborhood commuting matrices. And now “digital exhaust” extracted from the web and smartphones offers a glimpse of behavior not even measured in traditional data sources. Dave Donaldson’s keynote address on “The benefits of new data for measuring the benefits of new transportation infrastructure” at the Urban Economics Association meetings in October highlighted a number of such exciting developments (ship-level port flows, ride-level taxi data, credit-card transactions, etc).

But finer and finer data are not a free lunch. Big datasets bring computational burdens, of course, but more importantly our theoretical tools need to keep up with the data we’re leveraging. Most models of the spatial distribution of economic activity assume that the number of people per place is reasonably large. For example, theoretical results describing space as continuous formally assume a “regular” geography so that every location has positive population. But the US isn’t regular, in that it has plenty of “empty” land: more than 80% of the US population lives on only 3% of its land area. Conventional estimation procedures aren’t necessarily designed for sparse data sets. It’s an open question how well these tools will do when applied to empirical settings that don’t quite satisfy their assumptions.

Felix Tintelnot and I examine one aspect of this challenge in our new paper, “Spatial Economics for Granular Settings“. We look at commuting flows, which are described by a gravity equation in quantitative spatial models. It turns out that the empirical settings we often study are granular: the number of decision-makers is small relative to the number of economic outcomes. For example, there are 4.6 million possible residence-workplace pairings in New York City, but only 2.5 million people who live and work in the city. Applying the law of large numbers may not work well when a model has more parameters than people.

Felix and I introduce a model of a “granular” spatial economy. “Granular” just means that we assume that there are a finite number of individuals rather than an uncountably infinite continuum. This distinction may seem minor, but it turns out that estimated parameters and counterfactual predictions are pretty sensitive to how one handles the granular features of the data. We contrast the conventional approach and granular approach by examining these models’ predictions for changes in commuting flows associated with tract-level employment booms in New York City. When we regress observed changes on predicted changes, our granular model does pretty well (slope about one, intercept about zero). The calibrated-shares approach (trade folks may know this as “exact hat algebra“), which perfectly fits the pre-event data, does not do very well. In more than half of the 78 employment-boom events, its predicted changes are negatively correlated with the observed changes in commuting flows.

The calibrated-shares procedure’s failure to perform well out of sample despite perfectly fitting the in-sample observations may not surprise those who have played around with machine learning. The fundamental concern with applying a continuum model to a granular setting can be illustrated by the finite-sample properties of the multinomial distribution. Suppose that a lottery allocates I independently-and-identically-distributed balls across N urns. An econometrician wants to infer the probability that any ball i is allocated to urn n from observed data. With infinite balls, the observed share of balls in urn n would reveal this probability. In a finite sample, the realized share may differ greatly from the underlying probability. The figure below depicts this ratio for one urn when I balls are distributed across 10 urns uniformly. A procedure that equates observed shares and modeled probabilities needs this ratio to be one. As the histograms reveal, the realized ratio can be far from one even when there are two orders of magnitude more balls than urns. Unfortunately, in many empirical settings in which spatial models are calibrated to match observed shares, the number of balls (commuters) and the number of urns (residence-workplace pairs) are roughly the same. The red histogram suggests that shares and probabilities will often differ substantially in these settings.

Balls and 10 urns: Histogram of realized share divided by underlying probability

Granularity is also a reason for economists to be cautious about their counterfactual exercises. In a granular world, equilibrium outcomes depend in part of the idiosyncratic components of individuals’ choices. Thus, the confidence intervals reported for counterfactual outcomes ought to incorporate uncertainty due to granularity in addition to the usual statistical uncertainty that accompanies estimated parameter values.

See the paper for more details on the theoretical model, estimation procedure, and event-study results. We’re excited about the growing body of fine spatial data used to study economic outcomes for regions, cities, and neighborhoods. Our quantitative model is designed precisely for these applications.

# Do customs duties compound non-tariff trade costs? Not in the US

For mathematical convenience, economists often assume iceberg trade costs when doing quantitative work. When tackling questions of trade policy, analysts must distinguish trade costs from import taxes. For the same reason that multiplicative iceberg trade costs are tractable, in these exercises it is easiest to model trade costs as the product of non-policy trade costs and ad valorem tariffs. For example, when studying NAFTA, Caliendo and Parro (2015) use the following formulation:

Caliendo and Parro (REStud, 2015), equation (3)

This assumption’s modeling convenience is obvious, but do tariff duties actually compound other trade costs? The answer depends on the importing country. Here’s Amy Porges, a trade attorney, answering the query on Quora:

Tariff rates in most countries are levied on the basis of CIF value (and then the CIF value + duties is used as the basis for border adjustments for VAT or indirect taxes). CIF value, as Mik Neville explains, includes freight cost. As a result, a 5% tariff rate results in a higher total amount of tariffs on goods that have higher freight costs (e.g. are shipped from more distant countries).

The US is one of the few countries where tariffs are applied on the basis of FOB value. Why? Article I, section 9 of the US Constitution provides that “No Preference shall be given by any Regulation of Commerce or Revenue to the Ports of one State over those of another”, and this has been interpreted as requiring that the net tariff must be the same at every port. If a widget is loaded in Hamburg and shipped to NY, its CIF price will be different than if it were shipped to New Orleans or San Francisco. However the FOB price of the widget shipped from Hamburg will be the same regardless of destination.

Here’s a similar explanation from Neville Peterson LLP.

On page 460 of The Law and Policy of the World Trade Organization, we learn that Canada and Japan also take this approach.

Pursuant to Article 8.2, each Member is free either to include or to exclude from the customs value of imported goods: (1) the cost of transport to the port or place of importation; (2) loading, unloading, and handling charges associated with the transport to the port of place or importation; and (3) the cost of insurance. Note in this respect that most Members take the CIF price as the basis for determining the customs value, while Members such as the United States, Japan and Canada take the (lower) FOB price.

While multiplicative separability is a convenient modeling technique, in practice ad valorem tariff rates don’t multiply other trade costs for two of the three NAFTA members.

# Shift-share designs before Bartik (1991)

The phrase “Bartik (1991)” has become synonymous with the shift-share research designs employed by many economists to investigate a wide range of economic outcomes. As Baum-Snow and Ferreira (2015) describe, “one of the commonest uses of IV estimation in the urban and regional economics literature is to isolate sources of exogenous variation in local labor demand. The commonest instruments for doing so are attributed to Bartik (1991) and Blanchard and Katz (1992).”

The recent literature on the shift-share research design usually starts with Tim Bartik’s 1991 book, Who Benefits from State and Local Economic Development Policies?. Excluding citations of Roy (1951) and Jones (1971), Bartik (1991) is the oldest work cited in Adao, Kolesar, Morales (QJE 2019). The first sentence of Borusyak, Hull, and Jaravel’s abstract says “Many studies use shift-share (or “Bartik”) instruments, which average a set of shocks with exposure share weights.”

But shift-share analysis is much older. A quick search on Google Books turns up a bunch of titles from the 1970s and 1980s like “The Shift-share Technique of Economic Analysis: An Annotated Bibliography” and “Dynamic Shift‐Share Analysis“.

Why the focus on Bartik (1991)? Goldsmith-Pinkham, Sorkin, and Swift, whose paper’s title is “Bartik Instruments: What, When, Why and How”, provide some explanation:

The intellectual history of the Bartik instrument is complicated. The earliest use of a shift-share type decomposition we have found is Perloff (1957, Table 6), which shows that industrial structure predicts the level of income. Freeman (1980) is one of the earliest uses of a shift-share decomposition interpreted as an instrument: it uses the change in industry composition (rather than differential growth rates of industries) as an instrument for labor demand. What is distinctive about Bartik (1991) is that the book not only treats it as an instrument, but also, in the appendix, explicitly discusses the logic in terms of the national component of the growth rates.

I wonder what Tim Bartik would make of that last sentence. His 1991 book is freely available as a PDF from the Upjohn Institute. Here is his description of the instrumental variable in Appendix 4.2:

In this book, only one type of labor demand shifter is used to form instrumental variables2: the share effect from a shift-share analysis of each metropolitan area and year-to-year employment change.3 A shift-share analysis decomposes MSA growth into three components: a national growth component, which calculates what growth would have occurred if all industries in the MSA had grown at the all-industry national average; a share component, which calculates what extra growth would have occurred if each industry in the MSA had grown at that industry’s national average; and a shift component, which calculates the extra growth that occurs because industries grow at different rates locally than they do nationally…

The instrumental variables defined by equations (17) and (18) will differ across MSAs and time due to differences in the national economic performance during the time period of the export industries in which that MSA specializes. The national growth of an industry is a rough proxy for the change in national demand for its products. Thus, these instruments measure changes in national demand for the MSA’s export industries…

Back in Chapter 7, Bartik writes:

The Bradbury, Downs, and Small approach to measuring demand-induced growth is similar to the approach used in this book. Specifically, they used the growth in demand for each metropolitan area’s export industries to predict overall growth for the metropolitan area. That is, they used the share component of a shift-share analysis to predict overall growth.

Hence, endnote 3 of Appendix 4.2 on page 282:

This type of demand shock instrument was previously used in the Bradbury, Downs and Small (1982) book; I discovered their use of this instrument after I had already come up with my approach. Thus, I can only claim the originality of ignorance for my use of this type of instrument.

Tim once tweeted:

Researchers interested in “Bartik instrument” (which is not a name I coined!) might want to look at appendix 4.2, which explains WHY this is a good instrument for local labor demand. I sometimes sense that people cite my book’s instrument without having read this appendix.

Update (10am CT): In response to my query, Tim has posted a tweetstorm describing Bradbury, Downs, and Small (1982).

# What share of US manufacturing firms export?

What share of US manufacturing firms export? That’s a simple question. But my answer recently changed by quite a lot. While updating one of my class slides that is titled “very few firms export”, I noticed a pretty stark contrast between the old and new statistics I was displaying. In the table below, the 2002 numbers are from Table 2 of Bernard, Jensen, Redding, and Schott (JEP 2007), which reports that 18% of US manufacturing firms were exporters in 2002. The 2007 numbers are from Table 1 of Bernard, Jensen, Redding, and Schott (JEL 2018), which reports that 35% of US manufacturing firms were exporters in 2007.

NAICS Description Share of firms Exporting firm share Export sales share of exporters
2002 2007 2002 2007 2002 2007
311 Food Manufacturing 6.8 6.8 12 23 15 21
312 Beverage and Tobacco Product 0.7 0.9 23 30 7 30
313 Textile Mills 1.0 0.8 25 57 13 39
314 Textile Product Mills 1.9 2.7 12 19 12 12
315 Apparel Manufacturing 3.2 3.6 8 22 14 16
316 Leather and Allied Product 0.4 0.3 24 56 13 19
321 Wood Product Manufacturing 5.5 4.8 8 21 19 09
322 Paper Manufacturing 1.4 1.5 24 48 9 06
323 Printing and Related Support 11.9 11.1 5 15 14 10
324 Petroleum and Coal Products 0.4 0.5 18 34 12 13
325 Chemical Manufacturing 3.1 3.3 36 65 14 23
326 Plastics and Rubber Products 4.4 3.9 28 59 10 11
327 Nonmetallic Mineral Product 4.0 4.3 9 19 12 09
331 Primary Metal Manufacturing 1.5 1.5 30 58 10 31
332 Fabricated Metal Product 19.9 20.6 14 30 12 09
333 Machinery Manufacturing 9.0 8.7 33 61 16 15
334 Computer and Electronic Product 4.5 3.9 38 75 21 28
335 Electrical Equipment, Appliance 1.7 1.7 38 70 13 47
336 Transportation Equipment 3.4 3.4 28 57 13 16
337 Furniture and Related Product 6.4 6.5 7 16 10 14
339 Miscellaneous Manufacturing 9.1 9.3 2 32 15 16
Aggregate manufacturing 100 100 18 35 14 17

Did a huge shift occur between 2002 and 2007? No. The difference between these two tables is due to a change in the data source used to identify whether a firm exports. In their 2007 JEP article, BJRS used a question about export sales in the Census of Manufactures (CM). In their 2018 JEL article, BJRS used customs records from the Longitudinal Firm Trade Transactions database (LFTTD) that they built. Footnote 23 of the latter article notes that “the customs records from LFTTD imply that exporting is more prevalent than would be concluded based on the export question in the Census of Manufactures.”

This is a bit of an understatement: only about half of firms that export in customs records say that they export when asked about it in the Census of Manufactures! [This comparison is inexact because the share of exporting firms may have really increased from 2002 to 2007, but BJRS (2018) say that they “find a relatively similar pattern of results for 2007 as for 2002” when they use the CM question for both years.] The typical three-digit NAICS industry has the share of firms that export roughly double when using customs data rather than the Census of Manufactures survey response. Who knows what happened in “Miscellaneous Manufacturing” (NAICS 339), which had 2% in the 2002 CM and 35% in the 2007 LFTTD.

I presume that the customs records are more reliable than the CM question. More firms are exporters than I previously thought!

# On “hat algebra”

This post is about “hat algebra” in international trade theory. Non-economists won’t find it interesting.

### What is “hat algebra”?

Alan Deardorff’s Glossary of International Economics defines “hat algebra” as

The Jones (1965) technique for comparative static analysis in trade models. Totally differentiating a model in logarithms of variables yields a linear system relating small proportional changes (denoted by carats (^), or “hats”) via elasticities and shares. (As published it used *, not ^, due to typographical constraints.)

The Jones and Neary (1980) handbook chapter calls it a circumflex, not a hat, when explaining its use in proving the Stolper-Samuelson theorem:

a given proportional change in commodity prices gives rise to a greater proportional change in factor prices, such that one factor price unambiguously rises and the other falls relative to both commodity prices… the changes in the unit cost and hence in the price of each commodity must be a weighted average of the changes in the two factor prices (where the weights are the distributive shares of the two factors in the sector concerned and a circumflex denotes a proportional change)… Since each commodity price change is bounded by the changes in both factor prices, the Stolper-Samuelson theorem follows immediately.

I’m not sure when “hat algebra” entered the lexicon, but by 1983 Brecher and Feenstra were writing “Eq. (20) may be obtained directly from the familiar ‘hat’ algebra of Jones (1965)”.

### What is “exact hat algebra”?

Nowadays, trade economists utter the phrase “exact hat algebra” a lot. What do they mean? Dekle, Eaton, and Kortum (2008) describe a procedure:

Rather than estimating such a model in terms of levels, we specify the model in terms of changes from the current equilibrium. This approach allows us to calibrate the model from existing data on production and trade shares. We thereby finesse having to assemble proxies for bilateral resistance (for example, distance, common language, etc.) or inferring parameters of technology.

Here’s a simple example of the approach. Let’s do a trade-cost counterfactual in an Armington model with labor endowment $L$, productivity shifter $\chi$, trade costs $\tau$, and trade elasticity $\epsilon$. The endogenous variables are wage $w$, income $Y = w \cdot L$, and trade flows $X_{ij}$. The two relevant equations are the market-clearing condition and the gravity equation.

Suppose trade costs change from $\tau_{ij}$ to $\tau'_{ij}$, a shock $\hat{\tau}_{ij} \equiv \frac{\tau'_{ij}}{\tau_{ij}}$. By assumption, $\hat{\chi}=\hat{L}=1$. We’ll solve for the endogenous variables $\hat{\lambda}_{ij}, \hat{X}_{ij}$ and $\hat{w}_{i}$. Define “sales shares” by $\gamma_{ij}\equiv\frac{X_{ij}}{Y_{i}}$. Algebraic manipulations deliver a “hat form” of the market-clearing condition.

Similarly, let’s obtain “”hat form” of the gravity equation.

Combining equations (1.1) and (1.2) under the assumptions that $\hat{Y}_{i}=\hat{X}_{i}$ and $\hat{\chi}=\hat{L}=1$, we obtain a system of equations characterizing an equilibrium $\hat{w}_i$ as a function of trade-cost shocks $\hat{\tau}_{ij}$, initial equilibrium shares $\lambda_{ij}$, and $\gamma_{ij}$, and the trade elasticity $\epsilon$:

If we use data to pin down $\epsilon, \lambda_{ij}$, and $\gamma_{ij}$, then we can feed in trade-cost shocks $\hat{\tau}$ and solve for $\hat{w}$ to compute the predicted responses of $\lambda'_{ij}$.

Why is this “exact hat algebra”? When introducing material like that above, Costinot and Rodriguez-Clare (2014) say:

We refer to this approach popularized by Dekle et al. (2008) as “exact hat algebra.”… One can think of this approach as an “exact” version of Jones’s hat algebra for reasons that will be clear in a moment.

### What is “calibrated share form”?

Dekle, Eaton, and Kortum (AERPP 2007, p.353-354; IMF Staff Papers 2008, p.522-527) derive the “exact hat algebra” results without reference to any prior work. Presumably, Dekle, Eaton, and Kortum independently derived their approach without realizing a connection to techniques used previously in the computable general equilibrium (CGE) literature. The CGE folks call it “calibrated share form”, as noted by Ralph Ossa and Dave Donaldson.

A 1995 note by Thomas Rutherford outlines the procedure:

In most large-scale applied general equilibrium models, we have many function parameters to specify with relative ly few observations. The conventional approach is to calibrate functional parameters to a single benchmark equilibrium… Calibration formulae for CES functions are messy and difficult to remember. Consequently, the specification of function coefficients is complicated and error-prone. For applied work using calibrated functions, it is much easier to use the “calibrated share form” of the CES function. In the calibrated form, the cost and demand functions explicitly incorporate

• benchmark factor demands
• benchmark factor prices
• the elasticity of substitution
• benchmark cost
• benchmark output
• benchmark value shares

Rutherford shows that the CES production function $y(K,L) = \gamma \left(\alpha K^{\rho} + (1-\alpha)L^{\rho}\right)^{1/\rho}$ can be calibrated relative to a benchmark with output $\bar{y}$, capital $\bar{K}$, and labor $\bar{L}$ as $y = \bar{y} \left[\theta \left(\frac{K}{\bar{K}}\right)^{\rho} + (1-\theta)\left(\frac{L}{\bar{L}}\right)^{\rho}\right]^{1/\rho}$, where $\theta$ is the capital share of factor income. If we introduce “hat notation” with $\hat{y} = y/\bar{y}$, we get $\hat{y} = \left[\theta \hat{K}^{\rho} + (1-\theta)\hat{L}^{\rho}\right]^{1/\rho}$. Similar manipulations of the rest of the equations in the model delivers a means of computing counterfactuals in the CGE setting.

# What economic activities are “tradable”?

I’ve had a couple conversations with graduate students in recent months about classifying industries or occupations by their tradability, so here’s a blog post reviewing some of the relevant literature.

A number of papers emphasize predictions that differ for tradable and non-tradable activities. Perhaps the most famous is Atif Mian and Amir Sufi’s Econometrica article showing that counties with a larger decline in housing net worth experienced a larger decline in non-tradable employment.

Mian and Sufi define industries’ tradability by two different means, one yielding a discrete measure and the other continuous variation:

The first method defines retail- and restaurant-related industries as non-tradable, and industries that show up in global trade data as tradable. Our second method is based on the idea that industries that rely on national demand will tend to be geographically concentrated, while industries relying on local demand will be more uniformly distributed. An industry’s geographical concentration index across the country therefore serves as an index of “tradability.”

Inferring tradability is hard. Since surveys of domestic transactions like the Commodity Flow Survey don’t gather data on the services sector, measures like “average shipment distance by industry” (Table 5a of the 2012 CFS) are only available for manufacturing, mining, and agricultural industries. Antoine Gervais and Brad Jensen have also pursued the idea of using industries’ geography concentration to reveal their tradability, allowing them to compare the level of trade costs in manufacturing and services. One shortcoming of this strategy is that the geographic concentration of economic activity likely reflects both sectoral variation in tradability and sectoral variation in the strength of agglomeration forces. That may be one reason that Mian and Sufi discretize the concentration measure, categorizing “the top and bottom quartile of industries by geographical concentration as tradable and non-tradable, respectively.”

We might also want to speak to the tradability of various occupations. Ariel Burstein, Gordon Hanson, Lin Tian, and Jonathan Vogel’s recent paper on the labor-market consequences of immigration varying with occupations’ tradability is a nice example. They use “the Blinder and Krueger (2013) measure of offshorability’, which is based on professional coders’ assessments of the ease with which each occupation could be offshored” (p.20). When they look at industries (Appendix G), they use an approach similar to that of Mian and Sufi.

Are there other measure of tradability in the literature?

# Where are the jobs? Don’t look too closely

Robert Manduca, a Harvard sociology PhD student, has put together a nice visualization of employment data that he titled “Where Are the Jobs?” It’s a great map, modeled after the very popular dot map of US residents by ethnicity. The underlying data come from the Longitudinal Employer-Household Dynamics (LEHD) program, which is a fantastic resource for economics researchers.

Since every job is represented by a distinct dot, it’s very tempting to zoom in and look at the micro detail of the employment geography. Vox’s Matt Yglesias explored the map by highlighting and contrasting places like Chicago and Silicon Valley. Emily Badger similarly marveled at the incredible detail.

Unfortunately, at this super-fine geographical resolution, some of the data-collection details start to matter. The LEHD is based on state unemployment insurance (UI) program records and therefore depends on how state offices reporting the data assign employees to business locations. When an employer operates multiple establishments (an establishment is “a single physical location where business transactions take place or services are performed”), state UI records don’t identify the establishment-level geography:

A primary objective of the QWI is to provide employment, job and worker flows, and wage measures at a very detailed levels of geography (place-of-work) and industry. The structure of the administrative data received by LEHD from state partners, however, poses a challenge to achieving this goal. QWI measures are primarily based on the processing of UI wage records which report, with the exception of Minnesota, only the employing employer (SEIN) of workers… However, approximately 30 to 40 percent of state-level employment is concentrated in employers that operate more than one establishment in that state. For these multi-unit employers, the SEIN on workers’ wage records identifies the employing employer in the ES-202 data, but not the employing establishment… In order to impute establishment-level characteristics to job histories of multi-unit employers, non-ignorable missing data model with multiple imputation was developed.

These are challenging data constraints. I have little idea how to evaluate the imputation procedures. These things are necessarily imperfect. Let me just mention one outlier as a way of illustrating some limitations of the data underlying the dots.

Census block 360470009001004 (that’s a FIPS code; “36” is New York “36047” is Kings County, and so forth) is in Brooklyn, between Court St and Adams St and between Livingston St and Joralemon St. The Borough Hall metro station is on the northern edge of the block. (Find it on the Census Block maps here). A glance at Google Maps shows that this block is home to the Brooklyn Municipal Building, Brooklyn Law School, and a couple other buildings.

What’s special about census block 360470009001004 is that it supposedly hosted 174,000 jobs in 2010, according to the LEHD Origin-Destination Employment Statistics (ny_wac_S000_JT01_2010.csv). This caught my eye because it’s the highest level in New York and really, really high. The other ten census blocks contained in the same census tract (36047000900) have less than 15,000 jobs collectively. This would be a startling geographic discontinuity in employment density. The census block with the second highest level of employment in the entire state of New York has only 48,431 employees.

A glance at the Brooklyn Municipal Building shows that it’s big, but it sure doesn’t make it look like a place with 174,000 employees.

And other data sources that do report employment levels by establishment (rather than state employer identification number) show that there aren’t 174,000 jobs on this block. County Business Patterns, a data set that is gathered at the establishment level, reports that total paid employment in March 2010 in ZIP code 11201, which contains this census block and many others,  was only 52,261. Looking at industries, the LODES data report that 171,000 of the block’s 174,000 jobs in 2010 were in NAICS sector 61 (educational services). Meanwhile, County Business Patterns shows only 28,117 paid employees in NAICS 61 for all of Brooklyn (Kings County) in 2010. I don’t know the details of how the state UI records were reported or the geographic assignments were imputed, but clearly many jobs are being assigned to this census block, far more than could plausibly be actually at this geographic location.

So you need to be careful when you zoom in. Robert Manduca’s map happens to not be too bad in this regard, because he limits the geographic resolution such that you can’t really get down to the block level. If you look carefully at the image at the top of this post and orient yourself using the second image, you can spot the cluster of “healthcare, education, and government” jobs on this block near Borough Hall just below Columbus Park and Cadman Plaza Park, which are jobless areas. But with 171,000 dots on such a tiny area, it’s totally saturated, and its nature as a massive outlier isn’t really visible. In more sparsely populated parts of the country, where census blocks are physically larger areas, these sorts of problems might be visually evident.

“Where Are The Jobs?” is an awesome mapping effort. It reveals lots of interesting information; it is indeed “fascinating” and contains “incredible detail“. We can learn a lot from it. The caveat is that the underlying data, like every other data source on earth, have some assumptions and shortcomings that make them imperfect when you look very, very closely.

P.S. That second-highest-employment block in New York state? It’s 360470011001002, across the street from the block in question. With 45,199 jobs in NAICS sector 48-49, Transportation and Warehousing. But all of Kings County reported only 18,228 employees in NAICS 48 in 2010 in the County Business Patterns data.

# How not to estimate an elasticity

The Cato Institute’s Randal O’Toole claims to debunk a recent paper suggesting a “fundamental of road congestion”.

In support of the induced-demand claim, Mann cites research by economists Matthew Turner of the University of Toronto and Gilles Duranton of the University of Pennsylvania. “We found that there’s this perfect one-to-one relationship,” Mann quotes Turner as saying. Mann describes this relationship as, “If a city had increased its road capacity by 10 percent between 1980 and 1990, then the amount of driving in that city went up by 10 percent. If the amount of roads in the same city then went up by 11 percent between 1990 and 2000, the total number of miles driven also went up by 11 percent. It’s like the two figures were moving in perfect lockstep, changing at the same exact rate.” If this were true, then building more roads doesn’t make traffic worse, as the Wired headline claims; it just won’t make it any better.

However, this is simply not true. Nor is it what Duranton & Turner’s paper actually said. The paper compared daily kilometers of interstate highway driving with lane kilometers of interstates in the urbanized portions of 228 metropolitan areas. In the average metropolitan area, it found that between 1983 and 1993 lane miles grew by 32 percent while driving grew by 77 percent. Between 1993 and 2003, lane miles grew by 18 percent, and driving grew by 46 percent.

That’s hardly a “perfect one-to-one relationship.”

The paper also calculated the elasticities of driving in relationship to lane kilometers. An elasticity of 2 would mean a 10 percent increase in lane miles would correspond with a 20 percent growth in driving; an elasticity of 1 would mean that lane miles and driving would track closely together. The paper found that elasticities were very close to 1 with standard errors of around 0.05. Even though this is contradicted by the previously cited data showing that driving grew much faster than lane miles, this is the source of Turner’s “perfect one-to-one relationship.”

My prior belief is that results published in the American Economic Review are unlikely to be debunked by a couple of paragraphs in a blog post. In this case, it’s fairly straightforward to explain why the average growth rates of lane kilometers and vehicle-kilometers traveled are not informative about the elasticity.

The lane-kilometer elasticity of VKT describes how changes in VKT relate to changes in lane kilometers. O’Toole tries to say something about this relationship by noting the average value of each. But describing the average growth rates does not say whether cities that experienced faster growth in lane kilometers also experienced faster growth in vehicle-kilometers traveled. It’s entirely possible for both averages to be positive and the elasticity relating them to be negative! Here are a few lines of Stata code to generate an example in which the averages are 32% and 77%, while the elasticity is -1.

clear
set obs 228
gen delta_lane = .32 + rnormal(0,.2)
gen delta_VKT = (.77 +.32) - delta_lane + rnormal(0,.2)
twoway (scatter delta_VKT delta_lane) (lfit delta_VKT delta_lane), graphregion(color(white))


That yields a figure like this:

Having made this econometric point, one can grab the data used in the Duranton and Turner paper to  note the average values and appropriately estimate the elasticity, revealing no contradiction whatsoever between these two moments.

use "Duranton_Turner_AER_2010.dta", clear
gen delta_VKT = log(vmt_IHU_93) - log(vmt_IHU_83)
gen delta_lane = log(ln_km_IHU_93) - log(ln_km_IHU_83)
summ delta*
reg delta_VKT delta_lane
twoway (scatter delta_VKT delta_lane) (lfit delta_VKT delta_lane), graphregion(color(white))
`

Across MSAs, the average VKT change was a 61 log-point increase, while the average lane kilometers change was a 25 log-point increase. That’s a ratio greater than two, but the estimated elasticity is 0.955. Hence Matt saying that he and Gilles found a one-to-one relationship. Their paper deals with various types of roads and instrumenting to infer the causal relationship, but I don’t need to describe those issues here. I’ve written enough to demonstrate why O’Toole’s blog post does not debunk the Duranton-Turner findings.

# “Ricardian Productivity Differences and the Gains from Trade”

You’ll recall that Ralph Ossa emphasized sectoral heterogeneity in trade elasticities as one reason the ACR formula might understate the gains from trade. I haven’t read it yet, but this new NBER WP by Andrei Levchenko and Jing Zhang also emphasizes the importance of sectoral heterogeneity in thinking about this topic:

[T]he simpler formulas that do not use information on sectoral trade volumes understate the true gains from trade dramatically, often by more than two-thirds. The error in the formulas across countries is strongly negatively correlated to the strength of Ricardian comparative advantage: the one-sector formula-implied gains understate the true gains from trade by more in countries with greater dispersion in sectoral productivity. The model-based exercise thus reinforces the main result of the paper that accounting for sectoral heterogeneity in productivity is essential for a reliable assessment of the gains from trade.